Using the Binet’s Formula:

What is the 35th term of the Fibonacci Sequence?

Responses

A	8,965,4878,965,487

B	317,811317,811

C	9,227,4659,227,465

D	604,419

To find the 35th term of the Fibonacci sequence using Binet's Formula, we use the following equation:

\[
F(n) = \frac{ \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n }{\sqrt{5}}
\]

Where:
- \( F(n) \) is the nth Fibonacci number,
- \( n \) is the position in the sequence (in this case, \( n = 35 \)).

Let’s calculate the 35th term using this formula. I'll simplify the terms and compute the result.

\[
F(35) = \frac{ \left( \frac{1 + \sqrt{5}}{2} \right)^{35} - \left( \frac{1 - \sqrt{5}}{2} \right)^{35} }{\sqrt{5}}
\]

### Approximate calculation:

1. \( \frac{1 + \sqrt{5}}{2} \approx 1.618 \)
2. \( \frac{1 - \sqrt{5}}{2} \approx -0.618 \)

Now, using these values and computing:

\[
F(35) \approx \frac{1.618^{35} - (-0.618)^{35}}{\sqrt{5}}
\]

Let me compute the value.The 35th term of the Fibonacci sequence is approximately \( 9,227,465 \).

Thus, the correct response is **C**.

Would you like more details on how this was calculated, or do you have any questions?

Here are 5 related questions to consider:
1. How does Binet's formula approximate Fibonacci numbers?
2. What is the significance of the golden ratio \( \phi \) in Fibonacci calculations?
3. Can you use Binet’s formula for any position in the Fibonacci sequence?
4. How do rounding errors affect the precision of Binet's formula for large \( n \)?
5. What are other methods to calculate large Fibonacci numbers besides Binet's formula?

**Tip:** Binet's formula is an approximation, but it becomes highly accurate for larger \( n \) because the negative term \( \psi^n \) approaches 0.

Using Binet’s Formula: What is the 35th term of the Fibonacci Sequence?

A. 8,965,487
B. 317,811
C. 9,227,465
D. 604,419

Learn how to calculate the 35th term of the Fibonacci sequence using Binet's Formula. This problem involves algebraic concepts, the golden ratio, and Binet’s theorem. Suitable for students in Grades 9-12.

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